Modular surface of tri-focal Cassini curve ContourPlot3D missing feet - Mathematica Stack Exchange most recent 30 from mathematica.stackexchange.com 2019-10-19T19:16:01Z https://mathematica.stackexchange.com/feeds/question/44825 https://creativecommons.org/licenses/by-sa/4.0/rdf https://mathematica.stackexchange.com/q/44825 6 Modular surface of tri-focal Cassini curve ContourPlot3D missing feet Gary Palmer https://mathematica.stackexchange.com/users/1157 2014-03-26T20:45:44Z 2015-05-31T23:41:06Z <p>I am wondering why the following fails to cover the surface at points near \$k = 0\$.</p> <pre><code>c[z_] := (z + 1) (z - 1) (z + 1 + I); ContourPlot3D[Abs[c[x + I y]] == k^3, {x, -2.5, 2}, {y, -2, 2}, {k, 0, 1.75}, Background -&gt; White, AxesLabel -&gt; {"x", "y", "k"}] </code></pre> <p><img src="https://i.stack.imgur.com/AFiPd.png" alt="Mathematica graphics"></p> <p>Furthermore, when the left hand side is complex expanded and the equation is rearranged with \$2xy+k^3\$ on the right, the curve looks significantly different, with one large and one small foot, both touching at zero. Why is that?</p> https://mathematica.stackexchange.com/questions/44825/modular-surface-of-tri-focal-cassini-curve-contourplot3d-missing-feet/44830#44830 9 Answer by Rahul for Modular surface of tri-focal Cassini curve ContourPlot3D missing feet Rahul https://mathematica.stackexchange.com/users/484 2014-03-26T22:00:18Z 2014-03-26T22:32:29Z <p><code>ContourPlot3D</code> is not very good at resolving thin features, because it only knows that the feature exists when one of the sampling points happens to land inside it. In general, one thing you can do is to increase <code>PlotPoints</code>, which improves the plot but takes a very long time.</p> <pre><code>ContourPlot3D[ Abs[c[x + I y]] == k^3, {x, -2.5, 2}, {y, -2, 2}, {k, 0, 1.75}, Background -&gt; White, AxesLabel -&gt; {"x", "y", "k"}, PlotPoints -&gt; 20] </code></pre> <p><img src="https://i.stack.imgur.com/ZiZ1l.png" alt="enter image description here"></p> <p>In this particular case, though, your plot is equivalent to \$k = |c(x+iy)|^{1/3}\$, so you could just use <code>Plot3D</code> instead. This is much faster because it only has to sample the two-dimensional \$xy\$ plane rather than the three-dimensional \$xyk\$ space. Then you can afford to make <code>MaxRecursion</code> quite large and it's still really quick to plot.</p> <pre><code>Plot3D[Abs[c[x + I y]]^(1/3), {x, -2.5, 2}, {y, -2, 2}, PlotRange -&gt; {0, 1.75}, Background -&gt; White, AxesLabel -&gt; {"x", "y", "k"}, ClippingStyle -&gt; None, BoxRatios -&gt; 1, MeshFunctions -&gt; {#1 &amp;, #2 &amp;, #3 &amp;}, MaxRecursion -&gt; 5] </code></pre> <p><img src="https://i.stack.imgur.com/0aUYH.png" alt="enter image description here"></p> <p>(I've added a few options to make it look like your original plot.)</p>